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Bulletin de la SMF - Titles - 123 - pages 87-105

Titles123

Sur l'annulation de l'homologie du complexe de Koszul gradué
Marc Chardin
Bulletin de la Société mathématique de France 123, number 1 (1995), 87-105
Download this article : PS file / PDF file

Résumé :
Si $
\mathbb {K}
$ est le complexe de Koszul construit avec s polynômes homogènes à coefficients dans un corps k et r la codimension (ou hauteur) de la variété projective associée, on montre classiquement que les modules $H_{p}(
\mathbb {K}
 )$ sont nuls pour p>s-r et que $H_{s-r}(
\mathbb {K}
 )\not = 0$. Le complexe de Koszul est dans ce cas naturellement gradué et le calcul de la dimension du k-espace vectoriel $H_{p}(
\mathbb {K}
_{\nu })$ (partie homogène de degré $\nu $ de $H_{p}(
\mathbb {K}
 )$) est un problème élémentaire d'algèbre linéaire.

Nous fournissons dans cet article une borne sur $\nu $ à partir de laquelle $H_{s-r}(
\mathbb {K}
_{\nu })\not = 0$ (sauf dans le cas, simple à étudier, où la variété associée est vide). Indépendamment de son intérêt intrinsèque, cette borne ramène donc le calcul de la (co)dimension à un simple problème d'algèbre linéaire. Du point de vue de l'étude de la complexité du calcul de la dimension, notre résultat donne, dans le pire des cas, une borne sensiblement moins bonne que les meilleures connues (cf. [G-H]) ; en revanche notre borne tient compte de la géométrie de la variété sous-jacente.

Nous rappelons dans la première partie quelques résultats de base sur le complexe de Koszul gradué. La clef de la preuve du théorème central est une étude la plus fine possible des variétés définies par r polynômes « assez généraux » de l'idéal engendré par les polynômes de départ. En plus des théorèmes « classiques » de Macaulay et Bertini, j'utilise pour cela, inspiré par le travail d'Amoroso [A], les notions d'élément superficiel et de clôture intégrale d'un idéal, qui remontent aux travaux de Samuel-Zariski et Northcott-Rees. A partir de là, l'utilisation d'un théorème de Briançon-Skoda-Lipman-Teissier sur la clôture intégrale des idéaux [L-T], joint à notre estimation de la fonction de Hilbert [C], nous permet de conclure.

Abstract:
The central theorem of this paper is a result on the degree where the higher non-zero homology module of the Koszul complex constructed with homogeneous polynomials over a field becomes non trivial. This result has a straightforward corollary on the complexity of the determination of the dimension of a projective variety.

Let us state precisely our result about the Koszul complex. If $P_{1},\ldots ,P_{s}$ are homogeneous polynomials of $A=k[X_{0},\ldots ,X_{n}]$ and $d_{i}=\deg P_{i}$, the Koszul complex :

\begin{displaymath}
\mathbb {K}
:= 0\longrightarrow \wedge ^{s} A^{s} \stackrel ...
 ...ckrel {\mathrm {d} _{1}}{\longrightarrow } A\longrightarrow 0, \end{displaymath}

\begin{displaymath}
\mathrm {d} _{p}(e_{i_{1}} \wedge \cdots \wedge e_{i_{p}}):=...
 ...ots \wedge \widehat {e_{i_{k}}} \wedge \cdots \wedge e_{i_{p}} \end{displaymath}

(with the notation $A^{s}=e_{1}A\oplus \cdots \oplus e_{s}A$) is graded if we put $\deg (Pe_{i_{1}}\wedge \cdots \wedge e_{i_{p}})=\deg P+ d_{i_{1}}+\cdots + d _{i_{p}}$, and the differentials are of degree zero. If we consider the degree $\nu $ part of this complex, we obtain a complex of k-vector spaces of finite dimensions that we shall note $
\mathbb {K}
_{\nu }$.

It is well known that if the homogeneous ideal $I=(P_{1},\ldots ,P_{s})$ is different from A (i.e. if $\deg P_{i}\gt$ for all i), then $H_{p}(
\mathbb {K}
 )=0$ for all $p\gt s-\mathrm {ht}(I)$ and $H_{p}(
\mathbb {K}
 )\not = 0$ for $p=s-\mathrm {ht} (I)$, this is proved for instance in the book of Northcott [N, chap. 8, § 5, thm 6].

As $H_{p}(
\mathbb {K}
)=\bigoplus _{\nu }H_{p}(
\mathbb {K}
_{\nu })$ we can conclude that $H_{p}(
\mathbb {K}
_{\nu })=0$ for all $\nu $ if $p\gt s-\mathrm {ht}(I)$ and $H_{p}(
\mathbb {K}
_{\nu } )\not = 0$ for some $\nu $ if $p=s-\mathrm {ht} (I)$. Our result is an effective version of this last result, namely

Theorem: With the above notations and hypotheses, suppose for example that $ d _{1}\geq d_{2}\geq \cdots \geq d_{s}\gt$, and let us put $r=\mathrm {ht} (I)$ and $\pi _{r}=d_{1}\cdots d_{r}$, $\sigma _{r}=d_{1}+\cdots +d_{r}-r$. Then :

(a)
$H_{p}(
\mathbb {K}
_{\nu })=0$ for all $\nu $ if p>s-r ;

(b)
if $r\leq n$, $H_{s-r}(
\mathbb {K}
 ,\nu )\not = 0$ for all

\begin{displaymath}
\nu \geq \max \Bigl \{ \sigma _{r}+1,{{(n-r)\pi _{r}\sigma _...
 ...ver {2\deg I}}\Bigr \} +r d _{r+1}+ d _{r+2} +\cdots + d _{s}; \end{displaymath}

(c)
if r=n+1, there exists $\nu \leq d_{1}+\cdots +d_{s}-r$ such that $H_{s-r}(\nu )\not = 0$.

Let us remark that it is easy to determine if we are in the case (c), as in this case $I_{\nu }=A_{\nu }$ for all $\nu \gt\sigma _{r}$ (and reciprocally).

As $\deg I\geq 1$ we know that for $\nu =\max \{ \sigma _{r}+1,\frac 12(n-r) \pi _{r}\sigma _{r}\} +r\mathrm {d} _{r+1}+\mathrm {d} _{r+2} +\cdots +\mathrm {d} _{s}$, $H_{s-r}(
\mathbb {K}
_{\nu })=0$ if and only if $\mathrm {ht} (I)\not = r$. The determination of $H_{s-r}(
\mathbb {K}
_{\nu })$ is easy linear algebra over k, as the linear maps are defined by the formula above, and so the expression on the natural bases of the k-vector spaces $( \wedge ^{p}A^{s})_{\nu }$ is simple and the corresponding matrices are block matrices, each block is either zero or corresponds to the multiplication by one of the Pi's in a degree $\leq \nu $.

The idea for proving the theorem is to take r general linear combinations of the generators and to examine how deep they are close to the ideal on the components of maximal dimension of the variety defined by I, and what they define outside. Our results were greatly inspired by the work of Amoroso [A].

The first remark is that r linear combinations defines a regular sequence (this is due to Kronecker), and so, by a theorem of Macaulay, the associated ideal Ir is unmixed of height r. The second fact is that Ir is reduced (and even smooth) outside the support of I; this is consequence of a refined version of Bertini's theorem that can be found in the book of Jouanolou [J]. And the third property comes from the study of the blowing-up of the ideal I and says that for all prime ${
\mathfrak 
P}$ associated to I of height r, the integral closures of the localizations of I and Ir at ${
\mathfrak 
P}$ are the same (this is mostly contained in [Z-S, vol. 2, chap. VII, § 8] and [N-R]).

From that point, the proof is going on this way :

1.
By a classical lemma (proved e.g. in [N] in the way of proving that $H_{s-r}(
\mathbb {K}
 )\not = 0$) as Ir is given by a regular sequence, $H_{s-r}(
\mathbb {K}
_{\nu })$ is isomorphic to $H_{s}(\overline {
\mathbb {K}
}_{\nu +\mathrm {d} _{1}+\cdots +\mathrm {d} _{r}...
 ...athrm {d} _{r+1}+\cdots +\mathrm {d} _{s})} \mid \forall i,\ PP_{i}\in I_{r} \}$ (here $\overline {
\mathbb {K}
}$ denotes the Koszul complex constructed on the quotient ring A/Ir).

2.
From the second result quoted above, there exists two pure dimensional ideals I' and J of height r such that $I_{r}=I'\cap J$ with ${\rm Supp}(I')\subseteq {\rm Supp}(I)$, $J=\sqrt {J}$ and the primes associated to J are not in the support of I (it is also possible that Ir=I' in which case the proof is simpler and essentially skips the next step).

3.
The Hilbert function of Ir is easily expressible in terms of $d_{1},\ldots ,d_{r}$. Using the upper bound of [C] for the Hilbert function of the ideal J and comparing to the expression for Ir, we are able to prove that for $\nu \geq \max \{ \sigma _{r}+1, (n-r)\pi _{r}\sigma _{r}/(2\deg I)\}$, $(J/I_{r})_{\nu }\not = 0$.

4.
From a result of Briançon and Skoda on the integral closure of ideals, that was generalized by Lipman and Teissier [L-T], and the third property of general linear combinations quoted before, we deduce that $JI^{r}\subseteq I_{r}$ and the theorem follows.


ISSN : 0037-9484
Publié avec le concours de : Centre National de la Recherche Scientifique